Some Properties of a Recently Introduced Approach to Ordinal Regression
DOI:
https://doi.org/10.17713/ajs.v39i3.244Abstract
The statistical properties of a novel approach to ordinal regression which was only recently introduced in the literature are discussed. It is shown that for ordinal explanatory variables the approach is equivalent to isotonic regression, with some advantages when dealing with two-sided alternatives. For ordinal response variables the procedure behaves very differently and is asymptotically equivalent to a two-sample t-test between the extremecategories. A penalized version is introduced to improve power, and the procedure
is evaluated using Monte Carlo simulations. Finally the method is applied to microarray gene expression data on prostate cancer.
References
Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. (1972). Statistical Inference under Order Restrictions; The Theory and Application of Isotonic Regression. New York: Wiley.
Bartholomew, D. J. (1959a). A test of homogeneity for ordered alternatives. I. Biometrika, 46, 36-48.
Bartholomew, D. J. (1959b). A test of homogeneity for ordered alternatives. II. Biometrika, 46, 328-335.
Best, M. J., and Chakravarti, N. (1990). Active set algorithms for isotonic regression: a unifying framework. Mathematical Programming, 47, 425-439.
Bomze, I. (1998). On standard quadratic optimization problems. Journal of Global Optimization, 13, 369-387.
Chu, W., Ghahramani, Z., Falciani, F., and Wild, D. L. (2005). Biomarker discovery in microarray gene expression data with Gaussian processes. Bioinformatics, 21, 3385-3393.
Frommlet, F. (2008). Critical remarks on a novel approach to ordinal regression without latent variables (Tech. Rep. No. 208-06). ISDS.
Liu, I., and Agresti, A. (2005). The analysis of ordered categorical data: An overview and a survey of recent developments. Sociedad de Estadistica e Investigacion Operativa, 14, 1-73.
McCullagh, P. (1980). Regression models for ordinal data (with discussion). Journal of the Royal Statistical Society, Series B, 42, 109-142.
Robertson, T., Wright, F. T., and Dykstra, R. L. (1988). Order Restricted Statistical Inference. New York: Wiley.
Singh, D., Febbo, P., Ross, K., Jackson, D., Manola, J., Ladd, C., et al. (2002). Gene expression correlates of clinical prostate cancer behavior. Cancer Cell, 1, 203-209.
Tezel, G., Nagasaka, T., Iwahashi, N., N., N. A., Iwashita, T., Sakata, K., et al. (1999). Different nuclear/cytoplasmic distributions of RET finger protein in different cell types, 49, 881-886.
Torra, V., Domingo-Ferrer, J., Mateo-Sanz, J. M., and Ng, M. (2006). Regression for ordinal variables without underlying continuous variables. Information Sciences, 176, 465-474.
Downloads
Published
How to Cite
Issue
Section
License
The Austrian Journal of Statistics publish open access articles under the terms of the Creative Commons Attribution (CC BY) License.
The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed.
Copyright on any research article published by the Austrian Journal of Statistics is retained by the author(s). Authors grant the Austrian Journal of Statistics a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified.
Manuscripts should be unpublished and not be under consideration for publication elsewhere. By submitting an article, the author(s) certify that the article is their original work, that they have the right to submit the article for publication, and that they can grant the above license.